# Help in Inverse Laplace Transform in Circuit Analysis

Inverse Laplace Transform $$\frac{1}{s^2 + \sqrt{2}s + 1}$$

so what I did it changed the denominator to complete the square format which is (s+$$\frac{\sqrt{2}}{2})^2$$ + $$\frac{1}{2}$$ , then I can solve for s, it will make it as ((s+ $$\frac{\sqrt{2}}{2}$$) + $$\frac{\sqrt{2}}{2}j$$)((s+ $$\frac{\sqrt{2}}{2}$$) - $$\frac{\sqrt{2}}{2}j$$)

So now, according to my professor and the sheet of paper is to do Partial Fraction Decomposition of this which is absurd to me because of complex roots it has:

$$\frac{1}{s^2 + \sqrt{2}s + 1}$$

= $$\frac{1}{(s+\frac{\sqrt{2}}{2})^2 + \frac{1}{2}}$$

= $$\frac{1}{(s+\frac{\sqrt{2}}{2})^2 + \frac{1}{2}}$$

Partial Fraction of Complex root will be

= $$\frac{K}{(s+ \frac{\sqrt{2}}{2}) + \frac{\sqrt{2}}{2}j}$$ + $$\frac{K^*}{(s+ \frac{\sqrt{2}}{2}) - \frac{\sqrt{2}}{2}j}$$

right here I am stuck and don't know what to do.

To solve this issue, you need to be familiar with Euler's formula: $$\sin(wt) = \frac{e^{jwt}-e^{-jwt}}{2j}$$ You definitely started on the right track with this issue, only finding the right value for $$K$$ and $$K^*$$ Might be hard. So lets continue from there: $$K\left(s+\frac{1}{2}\sqrt{2}-\frac{1}{2}\sqrt{2}j\right)+K^*\left(s+\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{2}j\right) = 1$$ $$Ks+K^*s = 0 \rightarrow K = -K^*$$ $$K\left(\frac{1}{2}\sqrt{2}-\frac{1}{2}\sqrt{2}j\right)+K^*\left(\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{2}j\right) = 1$$ $$-K\sqrt{2}j = 1$$ $$K = \frac{1}{\sqrt{2}}j = \frac{1}{2}\sqrt{2}j \rightarrow K^*=-\frac{1}{2}\sqrt{2}j$$ Right, I suppose you know this inverse Laplace transform rule: $$\frac{1}{s-a} \rightarrow e^{at}$$ Let substitute the found expression in it: $$\frac{1}{2}\sqrt{2}je^{(-\frac{1}{2}\sqrt{2}-\frac{1}{2}\sqrt{2}j)t}-\frac{1}{2}\sqrt{2}je^{(-\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{2}j)t}$$ $$\frac{1}{2}\sqrt{2}je^{-\frac{1}{2}\sqrt{2}t}\left(e^{-\frac{1}{2}\sqrt{2}jt}-e^{\frac{1}{2}\sqrt{2}jt}\right)$$ Next, it might be useful to know that: $$j = \frac{1}{-j}$$ So substituting that in the equation leads to: $$\sqrt{2}e^{-\frac{1}{2}\sqrt{2}t}\left(\frac{e^{\frac{1}{2}\sqrt{2}jt}-e^{-\frac{1}{2}\sqrt{2}jt}}{2j}\right)$$ As you can see, Euler's formula neatly fits in here. resulting in the final answer: $$\sqrt{2}e^{-\frac{1}{2}\sqrt{2}t}\sin(\frac{1}{2}\sqrt{2}t)$$
This might be considered the hard way, explaining every step only using one of the most basic inverse Laplace transforms. However, for the future, I suggest you add these inverse transforms to your list as well: $$Ke^{at}\sin(bt) = \frac{Kb}{(s-a)^2+b^2}$$ $$Ke^{at}\cos(bt) = \frac{K(s-a)}{(s-a)^2+b^2}$$ As the damped periodic response is a very normal physical response, so expect to see a lot of those.
EDIT: to append to my earlier answer, I see you already had the correct formula at hand (albeit slightly more elaborate). The goal is to find $$K$$ and its complex conjugate $$K^*$$. As shown in your table. the magnitude of $$K$$ and its angle $$\angle K$$ are used to describe the magnitude of the (undamped) oscillation and the phase angle. These can be easily found as follows: $$|K| = \sqrt{K\cdot K^*}$$ $$\theta = \arctan\left(\frac{\mathfrak{Im}(K)}{\mathfrak{Re}(K)}\right)$$ Since in this case $$K$$ is pure imaginary, its easy to see that $$|K| = 0.5\sqrt{2}$$ and $$\theta = -0.5\pi$$.
• how you got |K| and $\theta$ ? – EM4 Aug 5 '20 at 16:15
• you forgot the square root for $|K|$ and my mistake for $\theta$ you need to use the atan2 function. you cannot divide by the real part of $K$ as it is 0. as for the atan2, its a more elaborate arctangent that completely describes the entire unit circle: en.wikipedia.org/wiki/Atan2 – Petrus1904 Aug 6 '20 at 17:59